Question

the graph shows the position s = f(t) of a car t hours after 5:00 p.m. relative to its starting point s = 0, where s is measured in miles. (a) describe the velocity of the car. specifically, when is it speeding up and when is it slowing down? (b) at approximately what time is the car traveling the fastest? the slowest? (c) what is the approximate maximum velocity of the car? the approximate minimum velocity? (a) when is the car speeding up? (simplify your answer. type your answer in interval notation. round to the nearest grid - line as needed. use a comma to separate answers as needed.)

Explanation:

Step1: Recall the relationship between position - velocity

Velocity \(v(t)\) is the derivative of position \(s = f(t)\), i.e., \(v(t)=f^\prime(t)\). The car is speeding up when \(v(t)\) and the acceleration \(a(t)=v^\prime(t)\) have the same sign. On a position - time graph, the slope of the tangent line to the curve \(s = f(t)\) gives the velocity. The car is speeding up when the slope of the tangent line to the \(s - t\) graph is increasing in magnitude (either getting steeper in the positive or negative direction).

Step2: Analyze the graph for speeding up

Looking at the \(s - t\) graph, we can estimate the intervals where the slope of the tangent line is increasing. By observing the curvature of the graph, we find the intervals where the graph is getting steeper.

Step3: Recall the relationship for maximum and minimum velocity

The maximum and minimum of the velocity function \(v(t)\) occur where the acceleration \(a(t)=v^\prime(t) = 0\), which means the slope of the velocity - time graph (derivative of \(v\)) is zero. On the position - time graph, this corresponds to the points where the concavity changes (inflection points) or where the tangent line has a horizontal tangent in the velocity - time graph (derivative of position).

Step4: Estimate maximum and minimum velocity from the graph

We look for the points on the \(s - t\) graph where the slope of the tangent line reaches its maximum and minimum values. The maximum velocity occurs when the slope of the \(s - t\) graph is the steepest (either positive or negative), and the minimum velocity occurs when the slope of the \(s - t\) graph is closest to zero.

(a) The car is speeding up when the slope of the position - time graph is increasing in magnitude. By observing the graph, the car is speeding up on the intervals \((0,2)\) and \((5,8)\) (approximate values based on the shape of the graph).
(b) The car is traveling the fastest when the slope of the position - time graph is the steepest. This appears to be around \(t = 2\) hours. The car is traveling the slowest when the slope of the position - time graph is closest to zero, which seems to be around \(t=5\) hours.
(c) To find the approximate maximum and minimum velocity, we estimate the steepest and flattest slopes of the tangent lines to the position - time graph. The approximate maximum velocity occurs around \(t = 2\) and has a value (by estimating the slope of the tangent line at \(t = 2\)) of about \(v_{max}\approx - 30\) miles per hour (negative because the position is decreasing). The approximate minimum velocity occurs around \(t = 5\) and has a value of about \(v_{min}\approx0\) miles per hour.

Answer:

(a) \((0,2),(5,8)\)
(b) Fastest at \(t = 2\) hours, slowest at \(t = 5\) hours
(c) Approximate maximum velocity: \(-30\) miles per hour (around \(t = 2\) hours), approximate minimum velocity: \(0\) miles per hour (around \(t = 5\) hours)